UNIFIED ANALYSIS OF TWO-HOP COOPERATIVE AMPLIFY-AND-FORWARD MULTI-RELAY NETWORKS

7/30/2019 UNIFIED ANALYSIS OF TWO-HOP COOPERATIVE AMPLIFY-AND-FORWARD MULTI-RELAY NETWORKS

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International Journal of Computer Networks & Communications (IJCNC) Vol.4, No.6, November 2012

DOI : 10.5121/ijcnc.2012.4601 1

UNIFIEDANALYSIS OFTWO-HOP COOPERATIVE

AMPLIFY-AND-FORWARD MULTI-RELAY

NETWORKS

Oluwatobi Olabiyi1

and Annamalai Annamalai2

Centre of Excellence for Communication Systems Technology Research

Department of Electrical and Computer Engineering

Prairie View A&M University, Texas [email protected] [email protected]

ABSTRACT

This article develops an extremely simple and tight closed-form approximation for the moment

generating function (MGF) of signal-to-noise ratio (SNR) for two-hop amplify-and-forward relayed

paths over generalized fading environments. The resulting expression facilitates efficient analysis of two-

hop cooperative amplify-and-forward (CAF) multi-relay networks over a myriad of stochastic channel

models (including mixed-fading scenarios where fading statistics of distinct links in the relayed path may

be from different family of distributions). The efficacy of our proposed MGF expression for computing

the average symbol error rate (ASER), outage probability, and the ergodic capacity (with limited

channel side-information among cooperating nodes) is also studied. Numerical results indicate that the

proposed MGF expression tightly approximates the exact MGF formulas and outperforms the existing

MGF of lower and upper bounds of the half-harmonic mean (HM) SNR, while overcoming the difficulties

associated in deriving an accurate MGF formula for the end-to-end SNR over generalized fading

channels. Further application of our new closed-form formula for the MGF of end-to-end SNR for

evaluating the average bit and/or packet error rate with adaptive discrete-rate modulation in CAF relay

networks is also discussed.

KEYWORDS

cooperative diversity, average symbol error probability, outage probability, ergodic capacity, moment

generating function approach

1. INTRODUCTION

The broadcast nature of wireless transmissions has enabled a new communication paradigm

known as cooperative communications wherein the source node communicates with the

destination node with the help of one or more relay nodes to harness the inherent spatial

diversity gain in wireless networks without requiring multiple transceivers at the destination

node. It is an active and growing field of research because this form of user cooperation

diversity has the ability to overcome the practical implementation issue of packing a largenumber of antenna elements on small-sized hand-held portable wireless devices and sensor

nodes, besides enabling the source node to tap into the available resources of local

neighbouring nodes to increase its throughput, range, reliability, and covertness.

Cooperative diversity can be broadly categorized as one of amplify-and-forward, decode-and-

forward, and compress-and-forward relaying strategies, each corresponding to differentprotocol implementations at the relay nodes [1]-[2]. Other variations cooperative diversity
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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strategies include opportunistic, incremental, variable-gain and fixed-gain (either blind or semi-blind) relaying that are based on the availability of channel side information (CSI) and the

number of nodes actively participating in information relaying [1]-[4]. In this article, we

primarily focus on variable-gain cooperative amplify-and-forward (CAF) relaying strategy

although the analysis may be extended to other categories and variations of cooperativerelaying strategies.

While numerous performance metrics of CAF relay networks have been considered in the

literature including ergodic/outage capacity, outage probability, and ASER (see [5]-[25] and

references therein), most results in the literature were restricted to either Rayleigh or

Nakagami-m channels, or the authors resort to asymptotic analysis, or develop performancebounds. In fact, determination of the exact performance of CAF multi-relay networks over a

generalized fading environment with independent but non-identically distributed (i.n.d) fading

statistics via an analytical approach is known to be a daunting task. This is attributed to thedifficulty in deriving the exact probability density function (PDF) or the moment generating

function (MGF) of the end-to-end signal-to-noise ratio (SNR). For instance, the exact MGF

expression for the desired SNR over i.n.d Nakagami-m derived recently in [14] involves triple

summation terms involving kth

derivative of the product of Whittaker functions withcomplicated arguments, which is not easily evaluated using a general computing platform,

besides being restrictive to positive integer fading index m. Other exact formulas (based on

the half-harmonic mean (HM) tight bound of exact end-to-end SNR) for the PDF or the MGF

of SNR in CAF relay networks can be found in [5]-[7] (for Rayleigh fading), [8] (for

Nakagami-m environment with independent and identically distributed (i.i.d) fading statistics)and [16]. Although the development in [16] is interesting and their MGF approach can be

applied to a wide range of fading environments, the resulting integral expressions are often too

complicated to compute or very time-consuming (due to the need to evaluate a nested two-fold

integral term with complicated arguments that includes infinite series in some cases). To

circumvent this difficulty, some authors have developed bounds for the half-harmonic mean

(HM) MGF of end-to-end SNR of CAF multi-relay networks in Rayleigh [6][10], Nakagami-m

[11][12] and Rice [13] fading environments. In [9], Ribeiro et. al. developed an asymptoticexpression for multi-relay CAF diversity system that employs BPSK modulation in Rayleigh

and Rice fading channels (although all their results were limited to only Rayleigh channels)

using an asymptotic analysis technique similar to that developed in [26] and [27] for non-

cooperative diversity systems. In [15], the asymptotic analysis result of [9] was extended to a

Nakagami-m fading channel.

In this article, we develop a new unified approximate MGF expression for the SNR of two-hoprelayed path which is then used to derive a tight approximate MGF of end-to-end SNR for

multi-relay networks. Unlike the contributions from related works found in the literature, our

closed-form MGF formula requires only the knowledge of the MGF of SNR of individual links,which makes it readily applicable to mixed fading and composite multipath/shadowing (e.g.,

Suzuki distribution, K-distribution, G-distribution, etc.) environments. The efficiency and

accuracy of our proposed solution is compared to existing closed-form and/or integralexpressions (when available) to demonstrate its utility and versatility. Several important

performance metrics of 2-hop CAF relay networks are considered such as average symbol error

rate (ASER), outage probability and ergodic capacity. In addition, our MGF expression may beexploited for efficient evaluation of ASER and/or average packet error rate (APER) with

discrete-rate adaptive modulation and/or computation of the average detection probability of

relay-assisted energy detector over generalized fading channel. Numerical results indicate that


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our MGF expression is much closer to the exact MGF compared to the widely used upper andlower bounds for the MGF of half harmonic mean SNR (e.g., [10]-[13]) despite its simplicity

and generality. While our unified MGF formula is slightly less accurate compared to the MGF

of the half-harmonic mean SNR (only available for specific fading scenarios) in [5]-[8] and

[16], it is still very close to the exact MGF expression derived in [14] and [18] (also availablefor only specific fading environments) while ensuring numerically stable and low

computational cost. In fact, our proposed solution is perhaps the only accurate closed-form

MGF expression that can effectively capture the independent and non-identical distributed

(i.n.d) fading statistics across distinct wireless links of a route path in a unified manner. It is

important to note that although the mathematical framework developed in [16] is applicable togeneralized fading channels, it is quite cumbersome and/or numerically unstable for computing

the MGF of end-to-end SNR in certain cases such as Rice and Nakagami-q environments

because it requires the evaluation of an integral whose integrand is a product of infinite seriescontaining Bessel functions with complicated arguments.

Moreover, for the specific case of ASER analysis of CAF relay networks, we express the final

ASER in closed-form (i.e., as a weighted sum of MGF of end-to-end SNR). This is

accomplished by using the second-order exponential approximation for the conditional errorprobability (CEP) of M-ary phase shift keying (MPSK) and/or M-ary quadrature amplitude

modulation (MQAM) digital modulation schemes (e.g., 2( | ) b bs

P ae ce + ) derived in [19].

The resulting unified ASER expressions are much more general, and more accurate over a

wide range of channel SNRs (especially at larger values of fading severity index than thecorresponding asymptotic ASER formulas presented in [9] and [15] while ensuring a low

computational cost for evaluating the desired ASER (since they are in closed-form). It is also

important to highlight that the simplicity of our final approximate ASER formula may facilitatefurther system level optimization tasks (e.g., optimal power assignment and/or relay placement

in CAF multi-relay networks) although such investigations are beyond the scope of this article.

The remainder of this paper is organized as follows. In Section II, we briefly review the system

model and discuss the key steps in our development of a tight approximation for the MGF ofend-to-end SNR. Several applications of our proposed MGF formula are discussed in Section

III along with selected numerical results to highlight its utility (e.g., comparisons with related

results in the literature) followed by some concluding remarks in Section IV.

2. TIGHT APPROXIMATION FOR THE MGF OF SNR IN CAF RELAY

NETWORKS

Consider a cooperative wireless network model that comprises of a source node S which

communicates with a destination node D via a direct-link and through Namplify-and-forward

relays,Ri, ,{1,2,...., }i N in two transmission phases. During the initial Phase I, Sbroadcasts asignal toD and to the relays Ri, where the channel fading coefficients between SandD, Sand

the i-th relay node Ri,Ri andD are denoted by ,s dh , ,s ih and ,i dh respectively. During the secondphase of cooperation, each of the Nrelays transmits the received signal after amplification via

orthogonal transmissions. If a maximum ratio combiner (MRC) is employed at the destination

nodeD to coherently combine all the signals received during these two transmission phases, the

effective end-to-end SNR is given by [1], [10]-[14],


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, ,

, ,

1 1, ,

N N

s i i d

T s d s d i

i is i i d c

= == + = +

+ + (1)where c is a constant (i.e., assumes the value of 1 for the exact end-to-end SNR and 0 for half-

harmonic mean 1 bound), 2, , sa b a b oh E N = corresponds to the instantaneous SNRs of linka-b,

sEdenotes the average symbol energy and 0N corresponds to the noise variance. Hence the

MGF of end-to-end SNR shown in (1) can be expressed as

,

1

( ) ( )

N

T s d i

i

s s

=

= (2)The MGF of the direct link

,( )

s ds

is a single channel and can be easily obtained in the

literature (e.g., [28, Table 2.2]). However, it is well-known in the literature that evaluation of

the MGF of SNR for the relayed path, ,

, ,

s i i d

i

s i i d c

=

+ +

is quite challenging and the exact MGF

formula (c = 1) only exists for Nakagami-m fading channel with integer fading severity index,

viz., [14, Eq. (7)]

, ,1 1 2( )

1 1

0 0 1 0

2( ) 1 2 ( , , ) ( , , , )

s i i d

i

m m k n lE

n k l q

n ls s C n k l n k l q

q

+ +

= = = =

+ + =

(3)

where

,

1 2 1 22

2 2 2 2, , 2 2 2 2

1 2

2 24 411 2

1 , ,1 1

1 1

( 1)( , , , )

2 22( 1)

s dt

n l n l n l n ls d s d

m k q

m k q m k q

t s

t t t t l dn k l q e W W

dt

+ + + + + +

+ +

+ + + +

= + +

+ + = ,

1 , , 2 , ,, ,s i s i i d i d m m = = ,( 2 )/2 ( 1 2 )/2

1 21

,!( )! !( 1)!

s dn l m l n k

s d

Cl k l n m n

=

, ma,b and ,a b are the Nakagami-m

fading index and average received SNR of wireless linka-b, respectively, and W(.,.) denotes

the Whittaker function [29]. Although Eq. (3) has only finite summation terms, it involves the

evaluation ofkth

derivatives of product of Whitaker functions which is not necessarily a trivial

task (i.e., one may have to resort to a suitable computing platform such as MAPLE software to

compute the above MGF using a symbolic differentiation tool). Due to this limitation, several

researchers have considered a more tractable MGF for half-harmonic mean SNRi

(i.e., c = 0)

which has been shown to be very accurate at moderate and high SNR [5]-[8]. Even in this case,

the obtained closed-form results are still limited to i.n.d Rayleigh [5]-[7] (seemingly differentexpressions but numerically same), i.i.d [8] and i.n.d Nakagami-m channels (for positive

integer fading index m) [14]. These results are summarized in (4)-(8) for readers convenience.

Rayleigh Fading: [4, Eq. (20)], [5, Eq. (7)], [6, Eq. (52)]

1Most prior work computes the PDF or MGF of

, , , ,( )i s i k d s i i d = + which becomes accurate for moderate and large SNR

values.


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( ) ( ), , 2 21 1

1 1

( ) 3 5 512 1 2 12 2 2 22

1, , 1

416

( ) 3, ; ; 2, ; ;( )3 ( )

s i i d

i

A s A sHM

A s A s

s i i d

s F FA sA s

+ + +

+ +

= + + +

(4)

where ( )2 1 .,.;.;.F is the Gauss Hypergeometric function [29], 1, , , ,

1 1 2,

s i i d s i i d

A = + +

2

, , , ,

1 1 2,

p i i d p i i d

A = +

( ) ( ){ }

2

2

4 ( ) ( ) ( )

2( ) 4 ( )( )

2

cos ( ) 4

( ) , Re( ) 4i

p f s f s f s

pf s p f sHM

a f s p

s s pf s p

+

=

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UNIFIED ANALYSIS OF TWO-HOP COOPERATIVE AMPLIFY-AND-FORWARD MULTI-RELAY NETWORKS

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